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A334164
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a(n) is the number of ON-cells in the completed n-th level of a triangular wedge in the hexagonal grid of A151723 (i.e., after 2^k >= n generations of the automaton in A151723 have been computed).
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1
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1, 2, 2, 4, 2, 6, 4, 8, 2, 10, 6, 10, 4, 14, 8, 16, 2, 18, 10, 16, 8, 20, 12, 22, 6, 26, 14, 22, 8, 30, 16, 32, 2, 34, 18, 28, 16, 34, 18, 32, 14, 40, 22, 34, 16, 42, 24, 44, 10, 50, 26, 40, 20, 48, 28, 50, 14, 58, 30, 46, 16, 62, 32, 64
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OFFSET
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1,2
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COMMENTS
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Conjecture 1: Except for a(2^n + 1) = 2, n >= 1, for odd-numbered completed levels a lower bound of the ratio of ON-cells to the length of the level is (2^n + 2)/(3*2^(n+1) + 1) with limit 1/6, determined by the subsequence of levels starting with: 13, 25, 49, 97, 193, 385, 769, 1537, 3073, ..., and the associated ON-cell counts: 4, 6, 10, 18, 34, 66, 130, 258, 514, ..., as listed in the second column of each of the two triangles below.
The ON-cell counts for the indices in each row define a line of slope 1/2. The formula for the indices of levels in row k >= 2 is L(k, i) = 1 + Sum_{j = 0, ..., i} 2^(k-j), 0 <= i <= k - 2, and the formula for the associated numbers of ON-cells is C(k, i) = 2 + Sum_{j = 1..i} 2^(k-1-j), 0 <= i <= k - 2:
Index of the level: L(k, i) number of ON-cells: C(k, i)
k\i 0 1 2 3 4 5 6 k/i 0 1 2 3 4 5 6
2: 5 2: 2
3: 9 13 3: 2 4
4: 17 25 29 4: 2 6 8
5: 33 49 57 61 5: 2 10 14 16
6: 65 97 113 121 125 6: 2 18 26 30 32
7: 129 193 225 241 249 253 7: 2 34 50 58 62 64
8: 257 385 449 481 497 505 509 8: 2 66 98 114 122 126 128
...
The pairs ( L(k, i), C(k, i) ), for 0 <= i <= k-2, define a line of slope 1/2 for each k >= 3.
For triangle L(k, i): column 0 is A000051(n), n >= 2; column 1 is A181565(n), n >= 3; column 2 is A083686(n), n >= 2; columns 3 is A195744(n), n >= 1; column 4 is A206371(n), n >= 2; column 5 is A196657(n), n >= 1; the bounding diagonal is A036563(n), n >= 3.
For triangle C(k, i): column 1 is A052548(n), n >= 1; column 2 is A164094(n), n >= 1.
Conjecture 2: In an even-numbered completed level 2*n the fraction of ON-cells is bounded below by (23 * 2^n - 24)/(2^(n+5) - 36) with limit 23/32, determined by the subsequence of levels starting with: 28, 92, 220, 476, 988, 2012, 4060, ... .
There are 16 numbers less than 1000 that do not occur as the number of ON-cells in a completed level through level 16384: 136, 164, 330, 334, 402, 444, 526, 570, 598, 604, 614, 714, 740, 822, 832, 878.
Sequence A334169 of even-numbered completed levels in which all cells are ON-cells is a subsequence of this sequence.
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LINKS
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MATHEMATICA
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(* a169781[] and support functions are defined in A169781 and create the list nTriangle *)
a334164[n_] := Module[{k, levels={}}, a169781[n]; For[k=1, k<=n, k++, AppendTo[levels, Count[nTriangle[[k]], 1] - 2]]; levels]/; (n>=3 && IntegerQ[Log[2, n]])
a334164[64] (* sequence data *)
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CROSSREFS
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Cf. A000051, A036563, A052548, A083686, A151723, A164094, A181565, A195744, A196657, A206371, A334169.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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